SIMPLE TRACIALLY $\mathcal {Z}$ -ABSORBING C*-ALGEBRAS

Abstract We define a notion of tracial $\mathcal {Z}$ -absorption for simple not necessarily unital C*-algebras, study it systematically and prove its permanence properties. This extends the notion defined by Hirshberg and Orovitz for unital C*-algebras. The Razak-Jacelon algebra, simple nonelementary C*-algebras with tracial rank zero, and simple purely infinite C*-algebras are tracially $\mathcal {Z}$ -absorbing. We obtain the first purely infinite examples of tracially $\mathcal {Z}$ -absorbing C*-algebras which are not $\mathcal {Z}$ -absorbing. We use techniques from reduced free products of von Neumann algebras to construct these examples. A stably finite example was given by Z. Niu and Q. Wang in 2021. We study the Cuntz semigroup of a simple tracially $\mathcal {Z}$ -absorbing C*-algebra and prove that it is almost unperforated and the algebra is weakly almost divisible.